The words limb, limn sound the same but have different meanings and spellings. Why do limb, limn sound the same even though they are completely different words?
The answer is simple: limb, limn are homophones of the English language.
One of the larger branches of a tree.
One of the jointed appendages of an animal, such as an arm, leg, wing, or flipper, used for locomotion or grasping.
An extension or a projecting part, as of a building or mountain range.
One that is considered to be an extension, member, or representative of a larger body or group.
To depict by painting or drawing. See Synonyms at represent.
Definitions from The American Heritage® Dictionary of the English Language, 4th Edition and Wordnik.
Homophones (literally "same sound") are usually defined as words that share the same pronunciation, regardless of how they are spelled.
If they are spelled the same then they are also homographs (and homonyms); if they are spelled differently then they are also heterographs (literally "different writing").